Truncated Log-concave Sampling with Reflective Hamiltonian Monte Carlo

TL;DR

This paper introduces Reflective Hamiltonian Monte Carlo (ReHMC), an efficient algorithm for sampling from log-concave distributions within convex bodies, with theoretical guarantees and practical high-dimensional performance improvements.

Contribution

The paper presents ReHMC, a novel HMC-based sampling method with proven mixing time bounds for truncated log-concave distributions and demonstrates its practical efficiency in high dimensions.

Findings

ReHMC outperforms Hit-and-Run and Coordinate-Hit-and-Run in sampling speed.

ReHMC effectively handles high-dimensional truncated sampling.

The implementation is open source and validated on various datasets.

Abstract

We introduce Reflective Hamiltonian Monte Carlo (ReHMC), an HMC-based algorithm, to sample from a log-concave distribution restricted to a convex body. We prove that, starting from a warm start, the walk mixes to a log-concave target distribution $\pi(x) \propto e^{-f(x)}$, where $f$ is $L$-smooth and $m$-strongly-convex, within accuracy $\varepsilon$ after $\widetilde O(\kappa d^2 \ell^2 \log (1 / \varepsilon))$ steps for a well-rounded convex body where $\kappa = L / m$ is the condition number of the negative log-density, $d$ is the dimension, $\ell$ is an upper bound on the number of reflections, and $\varepsilon$ is the accuracy parameter. We also developed an efficient open source implementation of ReHMC and we performed an experimental study on various high-dimensional data-sets. The experiments suggest that ReHMC outperfroms Hit-and-Run and Coordinate-Hit-and-Run regarding the time it needs to produce an independent sample and introduces practical truncated sampling in thousands of dimensions.